FFI-rapport 2011/00007
Channel sounding for acoustic communications: techniques and shallow-water examples
Paul van Walree
Norwegian Defence Research Establishment (FFI) 11 April, 2011
FFI-rapport 2011/00007 1146
P: ISBN 978-82-464-1901-5 E: ISBN 978-82-464-11902-2
Keywords
Kanalmålinger Kanalsimulering
Akustisk kommunikasjon
Approved by
Connie Elise Solberg Project Manager
Elling Tveit Director of Research
Jan Erik Torp Director
English summary
This report deals with channel soundings, which are measurements of the time-varying impulse response of a propagation medium. The treatment focuses on underwater acoustic channels, the characterization of which is becoming increasingly important to support developments in the fast growing field of underwater communications and networking. The benefits of channel sounding include increased understanding of channel physics and modem performance, validation of channel models, and support for the establishment of standard test channels. Channel models are of crucial importance in communication transceivers and channel simulators.
The report creates awareness of the risk of measurement errors related to properties of the probe signal, such as aliasing and delay-Doppler coupling. Channel parameters are defined and numerous shallow-water example soundings are reviewed. There are many conclusions, the most important one perhaps being that there is no typical or average acoustic channel. The variation in statistical properties, delay spread, and Doppler spread is immense. This variation is not only found between geographical areas and seasons, but also on smaller scales. For instance, the examples show that a wind burst or a passing ship can completely alter the scattering properties of a channel.
Doppler spectra are examined for wideband and narrowband waveforms. In agreement with our previous work, the basic shape of measured spectra appears to be well characterized by stretched and compressed exponentials. The spectral width increases with the frequency, something that should be kept in mind upon applying narrowband tools and models to broadband waveforms. An empirical relationship is established that reduces the stretched-exponential spectrum to a single parameter.
The present report is useful in several ways. It can be used as a guide for channel soundings at sea, including straightforward signal processing and computation of channel parameters. It also helps to recognize measurement errors and other pitfalls. Parameterization of Doppler spectra is useful for stochastic channel modeling. Most importantly, however, the collection of example soundings emphasizes the wide variety of acoustic propagation channels. It illustrates the challenge to devise communication systems that are efficient and robust to the environment, as well as channel simulators that faithfully mimic these environments. On the other hand, the set of channels in this report is selected so as to demonstrate the diversity as a function of area, season, weather, and local disturbances. One is unlikely to encounter all these channels for a given application or mission.
Sammendrag
Denne rapporten omhandler kanalmålinger, det vil si målinger av den tidsvarierende impulsrespon- sen til et propagasjonsmedium. Fokuset er på den undervannsakustiske kanalen. Karakterisering av slike kanaler er blitt stadig viktigere for å støtte utviklingen i det hurtig voksende feltet undervannskommunikasjon og nettverk. Kanalmålinger muliggjør øket forståelse av kanalfysikk og modemytelse, validering av kanalmodeller, og understøttelse for iverksetting av standard testkana- ler. Kanalmodeller er av stor betydning i kommunikasjonssendere, -mottakere og kanalsimulatorer.
Rapporten belyser risikoen for målefeil, som folding og forvrengning på grunn av kopling mellom tidsforsinkelse og Doppler, og relaterer disse til probesignalets egenskaper. Videre defineres kanalparametre, og en rekke eksempler på gruntvanns-kanaler gjennomgås. Det er mange konklusjoner, den viktigste er muligens at det ikke eksisterer noen typisk eller gjennomsnitts akustisk kanal. Variasjonen i statistiske egenskaper, tidsspredning og Dopplerspredning, er enorm.
Disse variasjonene finnes ikke bare mellom ulike geografiske områder og årstider, men også på mindre skalaer. For eksempel kan et vindkast eller et passerende skip fullstendig forandre spredeegenskapene til en kanal.
Dopplerspektra er studert for både bredbånds- og smalbåndsbølgeformer. Formen på målte spektra viser seg å kunne beskrives godt ved en strukket/komprimert eksponentiell funksjon. Dette er i samsvar med de tidligere resultatene våre. Den spektrale bredden øker med frekvens, noe en bør være oppmerksom på ved anvendelse av smalbåndsmetoder og modeller på bredbåndssignaler. En empirisk sammenheng er etablert, som reduserer antall parametre i det eksponentielle spektret til en enkelt parameter.
Rapporten kan benyttes på flere måter: Den kan brukes som en guide for å gjøre kanalmålinger i sjøen, inkludert basis signalbehandling og beregning av kanalparametre. Den kan også være til hjelp for å gjenkjenne målefeil og andre feller. Parametrisering av Doppler spektra er nyttig for stokastisk kanalmodellering. Men viktigst er det at samplingen av eksempler viser den store variasjonen som finnes i akustiske propagasjonskanaler. Eksemplene illustrerer utfordringen ved å konstruere kommunikasjonssystemer som er robuste og effektive med hensyn på det akustiske miljø, og kanalsimulatorer som troverdig reproduserer disse miljøene. På den annen side er eksemplene i denne rapporten valgt for å vise variasjonsbredden i akustiske kanaler som funksjon av område, årstid, værforhold, og lokale forstyrrelser: det er lite sannsynlig at en møter alle disse kanalene i en gitt anvendelse eller misjon.
Contents
1 Introduction 7
2 The acoustic channel 7
3 Probe signals 9
3.1 PRBS and LFM waveforms 9
3.2 Spectrum 10
3.3 Ambiguity function 10
4 Channel sounding 11
4.1 Time-varying impulse response 11
4.2 Delay-Doppler spread 12
4.3 Aliasing 14
4.3.1 Overspread channels 14
4.3.2 Temporal aliasing 15
4.3.3 Spectral aliasing 17
4.3.4 LMS channel estimation 17
4.4 Time-varying Doppler shifts 18
4.5 Definitions of delay spread and Doppler spread 20
4.6 Temporal coherence 22
5 Example channels 23
5.1 Channel A 25
5.2 Channel B 26
5.3 Channel C 27
5.4 Channel D 28
5.5 Channel E 29
5.6 Channel F 30
5.7 Channel G 31
5.8 Channel H 32
5.9 Channel I 33
5.10 Channel I.2 34
5.11 Channel J 35
5.12 Channel K 36
5.13 Channel L 37
5.14 Channel M 38
5.15 Summary 39
6 Implications for communications 41
6.1 Benign versus overspread channel 41
6.2 Performance prediction 42
6.3 Effect of a wind burst on a communication channel 43
6.4 Phase drift in a stationary channel 45
7 Parameterization of Doppler spectra 45
7.1 Stretched and compressed exponentials 45
7.2 Narrowband spectra 46
7.3 Relation between shape and width 49
8 Summary and conclusions 50
References 51
Appendix A Additional figures 54
1 Introduction
There is a growing interest in underwater acoustic telemetry. Digital underwater communications and networking are becoming increasingly important, with numerous applications emerging in environmental monitoring, exploration of the oceans, and military missions. In contrast to terrestrial radio communications, acoustic communications still rely on extensive field experiments for evaluation of physical-layer algorithms. There is thus an increasing need for acoustic channel simulators and definition of standard test channels. With a growing literature [1, 2] on modulation schemes and their performance in a particular environment, there is still a lack of methods to compare such schemes for realistic underwater channels. By contrast, papers published in the rapidly emerging field of underwater communication networks often compare several network protocols by simulation, e.g. [3].
In order to construct a realistic acoustic channel model or simulator, knowledge of the channel physics and statistics is required. Channel simulation based entirely on acoustic modeling is highly ambitious, and in practice one requiresin situchannel soundings, i.e., measurements of the time- varying impulse response, to validate models or to drive channel replay simulators [4, 5]. Channel characterization, sounding in particular, is at the basis of realistic channel models and improves understanding of system performance at sea. Existing characterizations investigate miscellaneous aspects of acoustic channels, e.g. [6, 7, 8, 9, 10, 11, 12, 13].
This report describes the technique of channel sounding, basic signal processing, and computation of channel parameters. The relevance for communication systems is evident throughout this report, but channel sounding can also be used just to study the physics of propagation. Techniques are generally applicable to propagation channels, whether it concerns underwater channels, radio channels, or otherwise. Parts of the description are possibly more relevant to the underwater environment, such as correction for time-varying Doppler shifts, delay-Doppler coupling, and aliasing in both delay and frequency. A section with example soundings illustrates the diversity of shallow-water channels and a number of intriguing propagation phenomena.
2 The acoustic channel
The underwater acoustic channel is quite possibly nature’s most unforgiving wireless communica- tion medium [14]. Absorption at high frequencies, and ship noise at low frequencies, limit the usable bandwidth to between, say, a few kilohertz and several tens of kilohertz, depending on the range.
Horizontal underwater channels are prone to multipath propagation due to refraction, reflection and scattering. The sound speedc≈1.5 km/s is low compared with the speed of light and may lead to channel delay spreads of tens or hundreds of milliseconds. In certain environments reverberation can be heard ringing for seconds and ultimately limits the performance of communication systems. The low speed of sound is also at the origin of significant Doppler effects, which can be subdivided in time-varying frequency shifts and momentaneous frequency spreading due to various mechanisms.
Both phenomena contribute to the Doppler variance of received communication signals, but require
different measures at the receiver. A channel that disperses the signal power in both delay and frequency is known as a doubly spread channel.
Signal fluctuations due to changes within the propagation medium, as opposed to transceiver motion, occur on various time scales [15]. Seasonal, diurnal, and tidal cycles may significantly alter the sound speed profile, and thereby the channel impulse response and propagation loss, but their time scales are very long compared with the duration of a typical communication packet. Such cycles are associated with seasonal (diurnal, tidal) performance variations of communication systems, but not with Doppler spreading and channel tracking at the physical layer. The main cause of frequency spreading that challenges acoustic communication receivers has to be scattering of sound by wind- generated waves [16, 17, 18, 12], where “main cause” refers to both frequency of occurrence and magnitude of the effect. In addition, clouds of air bubbles may form under the surface in the presence of breaking waves [19, 20]. Bubbles scatter and absorb sound, and may also modify the sound speed profile in the top few meters of the water column, thereby enhancing scattering by waves [21]. Other causes of time variability are as diverse as swell, wakes of passing ships, fish shoals, bubble screens due to rain showers, internal waves, a fresh-water front due to river discharge ... the list is long.
Currents affect the acoustic wavelength rather than the frequency and do not lead to Doppler shifts, unless they are (rapidly) time varying.
The variety of shallow-water communication channels renders it difficult to design physical-layer solutions that are robust to geographical area, weather conditions, and season. For the same reason it is challenging to design a channel simulator that faithfully mimics all environments, or even to find the most appropriate measurement scheme forin-situsoundings. The following, presumably incomplete overview sketches the diversity of channels that can be encountered. The channel may be characterized by correlated or uncorrelated scattering, by (quasi)stationary, cyclostationary, or non- stationary scattering. Shallow-water propagation channels range from stable monopath propagation to overspread, and from sparse to densely populated impulse responses. The most energetic arrival
Figure 2.1 Depiction of a shallow-water channel with reflection and scatterers.
may be at the start of the impulse response, at the very end, or somewhere in between. Doppler power spectra range from heavy-tailed to Gaussian, are symmetrical or asymmetrical, centered on zero frequency shift or offset. The Doppler spread may be essentially the same for all paths, for instance for signaling through a sound channel, or vary by orders of magnitude in channels featuring a mixture of specular and surface-reflected paths. Some channels may be characterized as being deterministic, while other ones exhibit Rayleigh, Rician, or K-distributed fading [22]. Section 5 illustrates many of the channel properties mentioned in this paragraph with example shallow-water soundings.
In addition to signal propagation comes ambient noise, from many and varied sources [23]. Noise can be colored and Gaussian when dominated by breaking waves, or impulsive and white when dominated by snapping shrimp or cavitating ship propellers. There are many noise sources with different properties, such as precipitation, marine mammals, cracking ice, sonar systems, offshore construction, platform self-noise. Underwater communication systems are often not limited by noise but by the channel itself. That is, the receiver output signal-to-interference-plus-noise ratio (SINR) is well below the input signal-to-noise ratio (SNR) and text-book bit-error-ratio (BER) curves are inapplicable. Channel estimation errors introduce an effective noise term at the receiver output that reflects self-interference, and this term may be much larger than the true noise term. In that case the noise characteristics are of secondary importance. Whenever an acoustic communication system becomes noise limited, the noise characteristics are obviously important. This report does not deal with noise, but focuses on channel sounding.
3 Probe signals
3.1 PRBS and LFM waveforms
Channel sounding requires transmission of judiciously selected probe signals. The choice depends on channel characteristics, which may or may not be known in advance, the type of signal processing for channel estimation, and properties of the probe signal itself. This report mainly focuses on channel estimation by replica correlation and considers two probe signals that have properties in common but also with differences. These are pseudorandom binary sequences (PRBSs) and linear frequency-modulated (LFM) chirp trains. Hyperbolic frequency-modulated chirps, a popular choice for detection applications, are deemed less suitable for channel sounding [24].
A PRBS is a repetition of a maximal-length bit sequencecm ∈ {−1,1}, modulated onto a binary phase-shift keyed waveform. The cyclic autocorrelation function of a maximal-length sequence with length M has a zero-lag value ofM, and−1 elsewhere. The autocorrelation function is no different for a train of LFM chirps, but in both cases the nice autocorrelation properties only apply to static channels. Doppler effects degrade these properties. Withfc denoting the center frequency, B the bandwidth,T the duration, andu(t)the bit pulse shape, a single “ping” is given by
p(t) = sin(2πfct)
M∑−1 m=0
cmu (
t− m MT
)
, (3.1)
p(t) = sin (
2π [(
fc −B 2
) t+ B
2Tt2 ])
, (3.2)
for the PRBS and LFM, respectively. In both cases the channel probe signal is constructed as a seamless concatenation ofN pings
s(t) =
N∑−1 n=0
p(t−nT). (3.3)
The pings are transmitted head to tail, without any pause in between, which is especially important for the PRBS. The LFM ping duration should be chosen such that it covers an integer number of carrier cycles.
3.2 Spectrum
When the sounding is performed with the intention to create archive files for channel simulators operating in replay mode [4, 5], the preferred spectrum of the waveforms is wideband and flatband.
Frequencies not covered by the probe signal cannot be simulated. Simply put, if a replay simulator filters aB = 8kHz (communication) signal with a channel obtained from aB = 5kHz probe signal, the filtered signal loses 3 kHz of bandwidth. If the probe signal has a flat spectrum over a frequency intervalB, the measured channel can be used without further adaptation to filter communication waveforms with a bandwidth ≤ B. If the spectrum of the probe signal is not flat its envelope must be deconvolved out of the channel estimate, which leads to noise amplification. Examples of suitable signals are PRBSs using a root-raised-cosine spectrum with a small roll-off factor, and unweighted chirps. For applications where sidelobes in delay are undesirable, frequency weighting can be applied. This can be done at the transmitter but also at reception, leaving the weighting option open. Transfer functions of equipment also affect channel soundings, but here it can be argued that spectral compensation is not required if the same instruments, e.g. acoustic modems, are also employed for communications.
Sections 3.3 and 4.2 show processing examples for PRBS and LFM probes using fc = 14kHz.
The PRBS is constructed with a bit rate of 8000 bps and a root-raised-cosine spectrum with roll-off factor 1/8. This yields a flat spectrum between 10.5 and 17.5 kHz and a smooth roll-off towards 9.5 and 18.5 kHz. The LFM spectrum is weighted with the same root-raised-cosine shape, such that the PRBS and the LFM waveforms have precisely the same spectral envelope and a−3dB bandwidth ofB = 8kHz between 10 and 18 kHz. The tracking periodT is varied to tune the delay-Doppler observation windowT×T−1to different channels. These probe signal parameters also apply to all soundings in Sec. 5 that use a carrier frequency of 14 kHz.
3.3 Ambiguity function
The PRBS and LFM ping types are characterized by their ambiguity functions [25, 26] in Fig. 3.1.
These plots give the correlation filter response, for a single ping, as a function of time delay and frequency shift (quantified for the center frequency). A constant Doppler shiftυ is applied to the signal, which differs from the Dopplerspreadchannels considered later.
Time delay (ms)
Frequency shift (Hz)
−6 −4 −2 0 2 4 6
−40
−20 0 20 40
Time delay (ms)
−6 −4 −2 0 2 4 6
−40
−20 0 20 40
−50
−40
−30
−20
−10 0
Figure 3.1 PRBS and LFM ambiguity functions, usingT = 256ms.
At zero frequency shift the response is the same for the PRBS and LFM, provided that the PRBS ping is embedded in its cyclic structure. However, elsewhere the surfaces are very different. The PRBS offers a high resolution in both delay and Doppler, but suffers from clutter at υ ̸= 0. The chirp has a tilted function. It has a much stronger response (peak filter output) than the PRBS for Doppler-shifted signals, but this Doppler insensitivity comes with a delay shift of ∆τ = υ× T /B. This phenomenon is known as delay-Doppler coupling, or range-Doppler coupling in ranging applications. The LFM additionally shows some broadening for large frequency shifts.
4 Channel sounding
4.1 Time-varying impulse response
The key objective of channel sounding is to measure/estimate the channel impulse responseh(τ, t) as a function of time delayτ and timet. Once an estimate ofh(τ, t)is available, channel parameters can be derived. To obtain a discrete-time estimate with the probe signals described in Sec. 3.1, a recorded signal is brought to complex baseband, (down)sampled at a rate fs, and filtered with a baseband replica of the transmit ping. This filter is known as a matched filter or correlation filter.
Individual impulse responses are cropped from the long filter output and stacked, so as to obtain a matrix ofN complex impulse responses
h=h(q, n) (4.1)
with corresponding time delaysτ(q) =q/fsand time instantst(n) =nT. The pulse shape ofhhas a spectrum that is the square of the spectrum of the probe signal. A root raised cosine becomes a full raised cosine. Accurate stacking requires an integer number of samples per ping, and is simply implemented in Matlab by the reshape command. Abundant literature is available on other channel estimation algorithms, which may work on very different waveforms. They differ in the method to obtainh(τ, t), and may yield different kinds of estimation errors, i.e., departures of the measured h(τ, t)from the true channel. Apart from these differences, all further processing is the same.
4.2 Delay-Doppler spread
A discrete Fourier transform ofhwith respect tongives the spreading function S(q, k) =F(h) =
N∑−1 n=0
h(q, n) exp
(−2πink N
)
, (4.2)
wherek ∈ [
−N2,−N2 + 1, ...,N2 −1]
corresponds to frequency shiftsυ(k) = k/(N T). Physical units are absent in this discrete-time representation; |S(q, k)|2 is a two-dimensional density that gives the relative distribution of signal power, or energy, over the delay-Doppler plane. A stochastic version known as the scattering function can be obtained by taking the expectation of the spreading function. Scattering functions are meaningful in the context of wide-sense stationary uncorrelated scattering (WSSUS) [27]. Examples in this report use the spreading function, which can be considered as a single realization of the scattering function, if indeed the channel is stationary.
Spreading and scattering functions are channel properties and differ from the ambiguity function, which is a property of the waveform.
Summation over delay yields an estimate of the Doppler power spectrum Pυ(k) =
Q∑−1 q=0
|S(q, k)|2 , (4.3)
whereQ=fsT, and summation over the frequency shift an estimate of the power delay profile Pτ(q) =
N∑−1 k=0
|S(q, k)|2 . (4.4)
The Doppler spectrum is a power spectral density that characterizes the distribution of received signal power as a function of frequency shift. The power delay profile gives the power distribution over time delay. Units are missing; this report shows dimensionless normalized spectra and profiles.
Since their ambiguity functions are the same at υ = 0, PRBS and LFM probes yield the same estimate of the power delay profile in static channels. This is illustrated in Fig. 4.1. The figure shows the power delay profile estimate for a synthetic channel with L = 6 multipath arrivals, spaced by 4 ms, whose amplitude and phase are time-invariant. Although there is no noise in the simulation, there is a floor in the power delay profile. This is due to the finite sidelobe levels of the probe signal autocorrelation function. The floor level is obtained from the time-bandwidth product of a ping, or equivalently the m-sequence gainM:−20 log10((M + 1−L)/L) =−32.4dB.
The behavior of the PRBS and LFM probes in a rapidly time-varying channel is illustrated in Fig. 4.2. Fading multipath arrivals are mimicked by a tapped delay line with six arrivals, each with a Gaussian Doppler power spectrum characterized by a Doppler spread of 2σ. The spacing between the arrivals is 4 ms, and they still have the same power density, i.e., amplitude in the true power delay profile. When the product of Doppler spread and probe periodT becomes of order 1, there is channel variability within a ping and the path amplitude drops. This drop is the same for the PRBS and LFM, but otherwise the power delay profiles are very different. Although Fig. 3.1 shows
Time delay (ms)
Power density (dB)
0 4 8 12 16 20 24 28 32
−42
−36
−30
−24
−18
−12
−6 0
Time delay (ms)
0 4 8 12 16 20 24 28 32
−42
−36
−30
−24
−18
−12
−6 0
Figure 4.1 Normalized power delay profiles obtained with PRBS (left) and LFM (right) probes for a static channel. The illustration usesT = 32 ms probe signals.
Time delay (ms)
Power density (dB)
0 4 8 12 16 20 24 28 32
−42
−36
−30
−24
−18
−12
−6 0
Time delay (ms)
0 4 8 12 16 20 24 28 32
−42
−36
−30
−24
−18
−12
−6 0
Figure 4.2 Power delay profiles obtained with PRBS (left) and LFM (right) probes for a channel with 2σ = 0, 1, 4, 16, 64, 256 Hz and a 4-ms path spacing. The illustration uses T = 32 ms probe signals.
the correlation filter response at a constant frequency shift, it qualitatively explains why PRBS and LFM probes respond differently when there are amplitude and phase fluctuations between pings and within a ping. The measured spreading function is a convolution of the true spreading function and the waveform ambiguity function. Rapid fading causes signal energy to be uniformly scattered in delay for the PRBS. The energy scatter is more localized for the LFM and broadens the fading arrivals. The choice of probe signal thus depends on what is considered more harmful, a loss of resolution in delay (chirp) or an increased interference-plus-noise floor (PRBS). It is presently not clear why the floor has become lower for the LFM probe, compared with Fig. 4.1.
4.3 Aliasing
The period T must be at least as long as the channel delay spread in order to cover the entire impulse response, and the probe rateT−1 must be high enough to track the time variability. When the product of delay spread and Doppler spread, known as the channel spread factor, exceeds unity, these demands cannot be met simultaneously and the channel is said to be overspread. Overspread channels unavoidably introduce estimation errors, such as aliasing with cyclic probes. Aliasing may also occur in underspread channels ifTis improperly tuned to the channel.
When the tracking period T is too short, the impulse response is aliased in delay and multipath arrivals spaced byTin the true impulse response add up at tap positions mod(τ, T)in the processing.
This is called temporal aliasing in this report. On the other hand, when the channel fading processes contain frequency components larger than 1/(2T), aliasing occurs in the frequency domain. This form of aliasing is sometimes referred to as temporal aliasing, but to avoid confusion with aliasing in delay it will be called spectral aliasing in this report. It is in the frequency domain that aliases appear, anyway. A doubly aliased measurement can be expressed via the spreading function as
S(q, k) =˜
Jτ
∑
j=0 Jυ
∑
j′=−Jυ
S(q+jQ, k+j′N), (4.5)
wherejruns over as many additional intervalsJτ ≥0as needed to collect all time-delayed signal energy, andj′over as many additional intervals2Jυas needed to collect all frequency-spread energy.
Time delay lacks a clear origin, unless the travel time from sender to receiver is precisely known, and the indexjcan be chosen to assume only positive values. There is a minimum travel time from sender to receiver, corresponding to a start of the received signal. Doppler spectra tend to be (near) symmetrical around zero frequency shift, and the indexj′ naturally assumes positive and negative values. Unlike the power delay profile, the Doppler spectrum has a clear origin.
4.3.1 Overspread channels
It is possible to get rid of temporal or spectral aliasing by tuning T, but not simultaneously if the channel is overspread. If one is only interested in the Doppler spectrum, a sufficiently short T ensuresJυ= 0. The measured Doppler spectrum becomes
P˜υ(k) =
Q∑−1 q=0
S(q, k)˜ 2
=
Q∑−1 q=0
Jτ
∑
j=0
S(q+jQ, k)
2
, (4.6)
which is not necessarily equal to the spectrum
Pυ(k) =
(Jτ+1)Q∑−1 q=0
|S(q, k)|2 (4.7)
derived from the true spreading function. In sparse channels there is a low probability of multipath overlap in the presence of temporal aliasing, and P˜υ will likely be correct. The same holds for uncorrelated scattering, as the power spectrum of a sum of uncorrelated processes equals the sum of the individual spectra.
On the other hand, if the interest is in the power delay profile a sufficiently longT ensuresJτ = 0.
Aliasing occurs in the Doppler spectrum and the properties of the probe signal enter the equation.
Spectral aliasing implies channel variations within a ping, and PRBSs and LFMs will distort the power delay profile as illustrated by Fig. 4.2.
The Doppler spread, due to reflection off moving scatterers and relative TX/RX motion, is normally well controlled in the sense that the velocities of the involved scatterers are bounded. This is different for the delay spread. Underwater environments may lead to channels that are characterized by a long reverberation tail, lasting hundreds of milliseconds. This reverberation is due to a three- dimensional scattering volume involving the seafloor, sea surface, and scatterers within the water column. The reverberation level decreases steadily with time delay, perceptibility ending only when it falls below the noise level. Temporal aliasing is unavoidable in such channels, if there is also Doppler to track, deceivingly lifting the noise floor in the power delay profile. Regardless of which form of aliasing is considered, and depending on the application, the channel sounding may still be useful if the aliased power is a sufficiently small fraction of the total.
4.3.2 Temporal aliasing
Figure 4.3 illustrates temporal aliasing. A stationary acoustic channel was probed withT = 128ms andT = 16ms waveforms. The resulting power delay profiles are normalized and synchronized so as to peak atτ = 0. The 128-ms measurement captures most of the time-dispersed power, but not 100% as there is still a decaying reverberation level at the end of the observation window. Notice that the measurement is cyclic, and the portion shown beforeτ = 0is the continuation of the tail at the right of the graph. The 16-ms probe covers too short a delay interval, and aliasing is observed.
First, the distinct arrivals atτ = 19andτ = 22ms in the 128-ms profile falsely appear atτ = 3 andτ = 6ms in the 16-ms data. Second, the long reverberation tail is multiply aliased and lifts the noise floor, filling gaps where the true profile has a low power density. Although the reverberation level is below –16 dB everywhere, the continuum of signal power density has a non-negligible integral. This is corroborated by the green curve in Fig. 4.3. This profile is obtained by assuming the 128-ms spreading function to be the true spreading function. An artificially aliased profile is computed via Eq. 4.5, usingJτ = 7andJυ= 0. The green profile is not separately normalized, but follows directly from the 128-ms data. There is a fairly good agreement between the measured and simulated 16-ms profiles. The huge increase of the interference-plus-noise floor is mostly accounted for by the aliased power, and perhaps for a small part by the lower processing gain of theT = 16ms waveform.
Time delay (ms)
Power density (dB)
0 10 20 30 40 50 60 70 80 90 100 110 120
−32
−28
−24
−20
−16
−12
−8
−4 0
128 ms Inferred 16 ms
Figure 4.3 Temporal aliasing: measured power delay profiles forT = 128 and 16 ms.
Frequency shift (Hz)
Power density (dB)
−32 −24 −16 −8 0 8 16 24 32
−28
−24
−20
−16
−12
−8
−4 0
16 ms Inferred 32 ms
Figure 4.4 Demonstration of spectral aliasing. Doppler spectra are shown for two probe signals, usingT = 16 and 32 ms.
4.3.3 Spectral aliasing
Spectral aliasing is shown in Fig. 4.4. A stationary acoustic channel was probed withT = 16ms andT = 32ms waveforms. The 16-ms probe has a wide Doppler observation window and captures the entire spectrum before it disappears below the noise floor. The 32-ms spectrum is identical to the 16-ms spectrum down to –16 dB, but deviates at its boundaries. In this particular case the Doppler spectrum is not symmetrical aboutυ= 0, and aliasing manifests itself notably with an upward bend at the left side of the 32-ms spectrum. Energy that falls out at the right side enters at the left side, and vice versa. The green spectrum is obtained by assuming the 16-ms spreading function to be the true spreading function. An artificially aliased spectrum is computed via Eq. 4.5, usingJτ = 0, Jυ = 1, and zero padding to obtain the required number of samples. The similarity of the green and red curves provides agreement between theory and practice, simultaneously supporting the claim of channel stationarity between the 16-ms and 32-ms measurements.
4.3.4 LMS channel estimation
Aliasing is a drawback of the cyclic structure of PRBSs or chirp ping trains, and can be avoided by using different waveforms and channel estimation algorithms. For example, a random phase- shift keyed symbol stream combined with the least mean squares (LMS) algorithm. Such a scheme allows channel updates at the symbol rate and has no limitation in time delay. Moreover, there is no need to decide on the tracking periodT before sea experiments. However, there exists a trade-off between channel length and tracking capability and there will be estimation errors. It is not expected
Time delay (ms)
Power density (dB)
0 10 20 30 40 50 60 70 80 90 100 110 120
−32
−28
−24
−20
−16
−12
−8
−4 0
128 ms 16 ms
Figure 4.5 Power delay profile estimates obtained by LMS channel estimation.
that alternative estimation methods will generally outperform the matched-filter estimator, but they may improve upon certain aspects such as aliasing.
Figure 4.5 illustrates this with power delay profiles obtained by LMS channel estimation. The algorithm is applied to a BPSK modulated random bit stream at 8000 bps that was transmitted in tandem with the probe signals of Fig. 4.3. Signal spectrum and SNR are the same. For comparison with the matched-filter estimates in Fig. 4.3 the estimation is performed with two channel vector lengths, corresponding to delay intervals of 128 and 16 ms. Note that the profiles in Fig. 4.5 are obtained from a single waveform, whereas the profiles in Fig. 4.3 come from two separate signals.
The 128-ms LMS estimate resembles the matched-filter result, but underestimates the power density of the delayed arrivals and the reverberation tail. In this case the reverberation originates from surface scattering, and is thus time varying. With the large number of channel taps required for a 128-ms delay coverage, the convergence time of the algorithm is too long to track rapid time variations. LMS channel estimation has difficulty with such channels, a conclusion that is based on more data than the single comparison between Fig. 4.3 and Fig. 4.5. LMS Doppler spectra tend to be narrower than their correlation-estimator counterparts. On the other hand, the 16-ms LMS estimate in Fig. 4.5 is more clean than its counterpart in Fig. 4.3 as there is no aliasing, but there is still an effective noise floor. Arrivals outside the extent of the channel vector act as noise. The LMS estimate for the 16-ms interval is better than in Fig. 4.3 whereas the 128-ms LMS profile is worse.
Obviously a 16-ms profile misses a large part of the impulse response, and the 128-ms matched-filter profile is the best overall result. A detailed comparison between the correlation estimator and other methods is beyond the scope of this report. All soundings in Sec. 5 use the correlation estimator.
4.4 Time-varying Doppler shifts
The WSSUS assumption does not always apply to underwater acoustic channels. There are many possible reasons, but one of the most prominent causes is TX/RX motion. When the transmitter and receiver move relative to each other, the signal experiences time compression or dilation. This manifests itself as a time-varying travel time and a twofold violation of WSSUS. The time dilation is to a large extent the same for all propagation paths, depending on the geometry, and introduces correlation in range rate and phase. Non-zero range rates cause multipath arrivals to drift between taps, in a tapped-delay-line representation, resulting in channel non-stationarity. Channel soundings reveal a smearing of the power delay profile and a broadening of the Doppler spectrum, even if the propagation medium itself is time-invariant.
The Doppler variance can receive significant contributions from TX/RX motion even if modems are deployed from an anchored ship or gateway buoy. Channel soundings meant to study the propagation medium itself require senders and transmitters in a frame on the seafloor, fixed to a quay, or similar. This is not always feasible, and the next best thing is to remove time-varying Doppler shifts (TVD) from the data by resampling operations. Fig. 4.6 shows an iterative procedure culminating in the elimination of kinematic effects from the measured impulse responseh(q, n).
The first step is the removal of a constant velocity v0 by resampling the received signal with a
Figure 4.6 Iterative procedure to eliminate constant and time-variable Doppler shifts from the soundingh(q, n).
resampling factor1−v0/cprior to the matched filter, where c = 1500 m/s is the nominal speed of sound and the sign convention forv0 such that a positive velocity corresponds to an increasing range. The value ofv0can be a rough guess, an automated estimate obtained with a bank of Doppler- shifted ping replicas, where a PRBS has an advantage over chirps, or be the result of a brute-force search. In the case of a stationary set-up it may be derived from known clock-frequency offsets, which result in apparent Doppler shifts. Its accuracy must be such that the bulk of the spectrum falls in the[−1/(2T),(1/2T)]regime after resampling. The next step is fine adjustment, for instance by shifting the center of gravity
υg =
N/2∑−1 k=−N/2
k
N TPυ(k)
N/2∑−1
k=−N/2
Pυ(k)
−1
, (4.8)
of the spectrum after the first resampling step, to zero frequency shift. The resampling factor that achieves this is given by1−V0/c, with
V0 =v0−υg
fcc . (4.9)
In case of apparent Doppler shifts due to clock frequency offsets of modem electronics, full correction is possible by resampling with a constant factor. However, the Doppler shift due to movement generally varies with time, which requires a third compensation step to remove TVD around the mean shift. To this end the unwrapped phaseθ(qj, n) ofh(qj, n) is upsampled from the channel tracking rate to the sampling frequency of the recorded data, and used to drive an interpolator that computes the signal at unevenly spaced timest′ =t−θ(qj, t)/(2πfc). The overall procedure is equivalent to a time-variable resampling factor
R(t) = 1−V0
c + 1 2πfc
dθ(qj, t)
dt (4.10)
applied to the recorded probe signal. A natural choice for θ(qj, t) is the phase of the most energetic arrival, although an average over multiple arrivals can also be considered. Notice that the intermediate step 2 in Fig. 4.6 is needed to obtain a good phase measurement for the third
step. Uncompensated Doppler causes multipath arrivals to gradually drift from one tap to the next, compromising the phase measurement for a given tap. In the presence of strong TX/RX acceleration even the conditionvg = 0is insufficient, with multipath arrivals noticeably wandering between taps.
See for instance Fig. 5.11. In such cases the third step of Fig. 4.6 may be repeated by accumulating phase measurements of successive iterations.
4.5 Definitions of delay spread and Doppler spread
The impulse response h(q, n) and its Fourier transform S(q, k) are complete descriptions of a measured channel. Derived scattering functions completely characterize the second-order statistics, but only for WSSUS channels. The power delay profile and Doppler spectrum present condensed information and do not allow channel reconstruction, unless it happens to be separable according to S(q, k) =√
Pτ(q)Pυ(k)/P, withP denoting the total signal power. A further loss of information occurs when the channel is reduced to only two parameters, known as the delay spread and the Doppler spread. These have limited significance and do not allow reconstruction of the power delay profile or Doppler spectrum, let alone the channel itself. An exception occurs when the profile or spectrum has a familiar shape. E.g., a Doppler spread of2σ= 1Hz completely describes a Gaussian power spectrum. Unfortunately real spectra are rarely Gaussian. Nonetheless parameterization of Doppler spectra appears to be possible, to some extent: see Sec. 7.
There are many definitions of delay and Doppler spread in circulation. The following definitions are for the delay spreadT; the Doppler spreadDis similarly defined from the Doppler power spectrum as the delay spread from the power delay profile. It is assumed that Pτ andPυ are normalized to unity.
The threshold delay spread is the difference between the longest and shortest delay where the profile equals or exceeds a given threshold valueX
Tthr=τ (
maxq {q|Pτ ≥X} )
−τ (
minq {q|Pτ ≥X} )
. (4.11)
The RMS delay spread is defined as
Trms=
Q∑−1
q=0
Pτ(q)[
τ(q)−τg]2
Q∑−1
q=0
Pτ(q)
−1
1/2
, (4.12)
whereτgdenotes the center of gravity
τg=
Q∑−1 q=0
τ(q)Pτ(q)
Q∑−1
q=0
Pτ(q)
−1
(4.13) of the power delay profile. The threshold and RMS definitions are two commonly used criteria, where RMS is said to be more relevant to the performance of communication systems [28]. A third, less common definition is physically intuitive and uses an energy (power) criterion. The normalized
accumulated power is computed E(q) =
∑q q′=0
Pτ(q′)
Q∑−1
q′=0
Pτ(q′)
−1
, (4.14)
and this function is used to find the border points maxq1
minq2 {q1, q2 |E(q2)−E(q1)≥Γ} , (4.15) of the shortest possible delay interval that captures a given fractionΓof the total signal power. The corresponding delay spreadτ(q2)−τ(q1)is denotedTen.
Note that the definitions are theoretical and that accurate measurement is not always feasible. Values are sensitive to noise and aliasing for all criteria, especially for smallX or large Γ. Cyclic delay shifts may additionally affect delay spread values. Channel estimation errors such as in Fig. 4.2 may have a big effect onTthr, whereasTrmsandTen will be fairly accurate for the LFM probe—but not for the PRBS. When accurate measurement is feasible, the definitions still have their particular strengths and weaknesses. A small change in multipath power can have a large effect on Tthr by shifting the amplitude of an early or late arrival just above or below the threshold value.Trms, when used to assess system performance, is overly sensitive to trailing arrivals with little power but a long delay [29]. Figure 4.7 illustrates the treacherous terrain. In this channel the cluster of delayed arrivals has an accumulated power that approximately equals the power of the precursor. There is a thin line between a delay spread of 0 and 400 ms using the −10 dB criterion, whereas a noise floor of−25dB over such a long stretch is high enough to affect the RMS and energy values. The 50%-energy delay spread obtained from the measured profile amounts to 200 ms, but should be
Time delay (ms)
Power density (dB)
−80 0 80 160 240 320 400 480 560
−30
−25
−20
−15
−10
−5 0
Figure 4.7 Peculiarly sparse power delay profile measured in the Baltic Sea.
much smaller for a noise-free profile. There is no substitute for inspection of a complete profile to see how it relates to the design of a communication scheme.
When the channel spread can be measured accurately, the energy criterion is attractive for system design and performance prediction. In order to be able to achieve a given performance, a receiver needs to have access to a given fraction of the received signal power in delay-Doppler space.
Signal energy that is not available, for instance because of a restricted equalizer length, acts as self-interference and lowers the signal-to-interference-plus-noise ratio (SINR).
4.6 Temporal coherence
A time-varying impulse response requires communication receivers to update their channel estimates in order to maintain good performance over time. A measure of the rate at which the channel changes is the coherence time, which is a time interval over which the coherence drops by some specified amount. Temporal coherence is addressed by the channel correlation function C(∆t), where∆t=t−t0 = (n−n0)T = ∆nT is a time interval relative to a reference timet0. In WSSUS theory, the correlation function is the inverse Fourier transform (IFT) of the Doppler power spectrum
Cυ(∆n) =F−1(Pυ(k)) (4.16)
=F−1
Q∑−1
q=0
S∗(q, k)S(q, k)
(4.17)
=F−1
Q∑−1
q=0
[F(h(q,∆n))]∗[F(h(q,∆n))]
(4.18)
=
Q∑−1 q=0
F−1
F
∑
j
h∗(q, j)h(q, j+ ∆n)
(4.19)
=
Q∑−1 q=0
∑
j
h∗(q, j)h(q, j+ ∆n) =Ca(∆n), (4.20) where the step between Eq. 4.18 and Eq. 4.19 uses the Autocorrelation (Wiener Khinchin) theorem.
Ca is the sum over all taps of the autocorrelation function with respect to t. Another estimate of temporal coherence from a measured channel is the normalized zero-lag cross-correlation with respect toτ
Cc(∆n) =
Q∑−1 q=0
h∗(q, n0)h(q, n) vu
utQ∑−1
q=0
h∗(q, n0)h(q, n0)
Q∑−1 q=0
h∗(q, n)h(q, n)
. (4.21)
The normalization is sometimes desirable and sometimes not. When the channel changes just by a scaling factor, a BPSK or QPSK receiver would not strictly require updates but a system employing
a higher-order constellation would. The zero-lag ingredient is also subject to discussion. When the entire impulse response shifts in delay, such as with time compression/dilation due to transceiver motion, some would say that the channel does not really change. However, an adaptive equalizer senses a change and needs to update its filter coefficients to deal with the delay shift.
The IFT correlation functionCυ, the autocorrelation functionCa, and the cross-correlation function Ccare all estimates and not necessarily the same for measured channels.CυandCaare equivalent on the WSSUS assumption, which requires an infinite observation period (infinite support forj). This condition is not met forin situsoundings, butCυ may be approached byCaif the latter is obtained via an unbiased estimate. Furthermore, stationarity is assumed andCυ andCa yield symmetrical correlation functions independent oft0. By contrast,Ccdoes not assume anything about the channel but just evaluates the similarity between successive impulse response measurements and a reference snapshot. It does not need to be symmetrical and may depend ont0 even in stationary channels.
It also suffers from an effective noise floor, because the cross-correlation of two random-noise vectors is not small unless the vectors are large. Ca is the expectation (time average) of Cc, if the normalization in Eq. 4.21 is omitted.
The example channels in the next section are illustrated with a comparison between the cross- correlation coherenceCcand the estimateCυobtained from the Doppler spectrum. Agreements and deviations provide clues as to the stationarity of the channel. Especially the presence or absence of Doppler spreading and time dilation due to TX/RX motion will prove to make a big difference.
The coherence time of a channel may be defined as the time over which a normalized correlation function drops from 1 to, say, 0.5, and will be denoted by C0.5. A single coherence time fails to adequately parameterize correlation functions of arbitrary shape, and the example channels of the next section demonstrate that the shape indeed varies considerably between channels.
5 Example channels
This section presents a cross section of channel soundings collected over the past five years. All examples concern horizontal shallow-water channels, the majority from northern European waters.
The collection is diverse and illustrates that there is no typical or average acoustic channel.
Thirteen channels are considered, labeled A through M. It is not feasible to describe sounding conditions and environments in detail, but Table 5.1 provides a concise overview. N is the number of pings used for the processed results; the actual number of transmitted pings is N + 2. The reason is that the first and last impulse response extracted from a PRBS are no part of its cyclic autocorrelation function. They are stripped off. An important column is the one that says whether the TX/RX deployment was static, with sender and receiver firmly mounted in frames on the seafloor (as in Fig. 2.1) or whether there was relative TX/RX motion. The SNR of the received probe signals is moderate to high, which is exactly what is needed for channel sounding. The purpose is to measure the time-varying impulse response, not noise.
Analysis results are shown in the form of figures with six panels, which showestimatesof:
• Channel,|h(q, n)|2, Eq. 4.1.
• Spreading function,|S(q, k)|2, Eq. 4.2.
• Doppler spectrum Eq. 4.3.
• Residual phase. This is the unwrapped phaseθ(qj, n)after resampling (if any). The phase measurement is shown for the main multipath arrivals, those which are within≈10 dB of the strongest path in the power delay profile. The black curve is the phase of the strongest path, weaker arrivals use shades of gray.
• Power delay profile Eq. 4.4.
• Correlation function. The magnitude of the functions Cυ(∆n) andCc(∆n), Eqs. 4.16 and 4.21. The reference impulse responseh(q, n0)is chosen halfway the probe signal.
All quantities plotted on a logarithmic scale are given in decibels relative to their maximum value.
Note that the soundings arepossiblyaliased, such that S˜is shown instead ofS, affecting also the other panels. This varies from case to case. The figures for channels A–M in the following sections use different color maps, so as to best highlight features of interest. For the same reason, the delay and frequency axes frequently zoom in on interesting regions. The summary in Tables 5.2 and 5.3 is always based on the fullT ×T−1observation window.
Ch. TX/RX Probe T N fc B Range Water SNR Month
(ms) (kHz) (kHz) (km) depth (dB)
(m)
A Static LFM 128 256 14 8 0.50 10 52 OCT
B Static PRBS 32 1024 14 8 0.75 5–16 29 MAY
C Static PRBS 32 1024 14 8 0.75 5–16 35 MAY
D Static Var. Var. Var. 14 8 1.15 20–80 15 OCT
E Static PRBS 32 1024 14 8 0.90 10–70 27 OCT
F Static PRBS 32 1024 14 8 0.90 10–70 26 OCT
G Static PRBS 32 1024 14 8 1.16 9–17 14 OCT
H Static LFM 128 256 14 8 1.16 9–17 31 OCT
I Moving PRBS 128 248 5 4 38 60–90 23 AUG
J Moving PRBS 128 248 5 4 21 250 11 SEP
K Moving PRBS 145 203 3.85 3.5 1.2 400 23 SEP
L Moving PRBS 102 290 15 10 1.5 100 30 JUN
M Moving PRBS 128 232 6 4 6.7 240 31 FEB
Table 5.1 Summary of experimental details for the example channels.
5.1 Channel A
An exceptionally benign communication channel was encountered in Horten’s natural bay. Propa- gation over 500 m at a 10-m water depth resulted in a spreading function that approaches a double Dirac function. There is a tiny bit of broadening in the delay profile, but only below –30 dB and close to the chief arrival. Similarly, the Doppler spectrum is a spike all the way down to –40 dB.
The frequency spread observed below this level is not due to noise, but to small fluctuations in signal amplitude and phase, visible as a ripple in the phase measurement. Possible causes of these fluctuations are channel physics, instrument jitter, transducer vibration by current. However, at
−50 dB such sidelobes are hardly important and this channel sustains high-rate communication without any form of channel equalization or tracking. Fig. A.1 reveals the noise floor.
To reveal the narrow Doppler spectrum with long-duration probe signals in a static channel, very precise resampling is necessary. Although the transmitter and receiver are bottom-mounted, there are offsets of their clock frequencies from the nominal value, resulting in an apparent Doppler shift.
A valueV0 =−0.1581m/s has been used (see Eq. 4.10) for Fig. 5.1; Fig. A.2 shows the sounding result forV0 =−0.16m/s. The phaseθ(τ = 0, t)is now subject to a drift of(∆V0/c)2πfcN T =
−3.7radians, enough to distort the Doppler spectrum and IFT correlation function. Total neglect of clock-frequency offsets (V0= 0) would seriously misrepresent the channel.
Time delay (ms)
Time (s)
−160 0 16 32 48 64 80 96 112 5
10 15 20 25 30
Channel
Time delay (ms)
Frequency shift (Hz)
−16 0 16 32 48 64 80 96 112
−3
−2
−1 0 1 2 3
Spreading function
Power density (dB)
Frequency shift (Hz)
−70 −60 −50 −40 −30 −20 −10 0
−3
−2
−1 0 1 2 3
Doppler spectrum
Time (s)
Phase (radians)
0 5 10 15 20 25 30
2.2 2.24 2.28 2.32 2.36
2.4 Residual phase
Time delay (ms)
Power density (dB)
−16 0 16 32 48 64 80 96 112
−70
−60
−50
−40
−30
−20
−10
0 Power delay profile
Time (s)
Coherence
0 5 10 15 20 25 30
0 0.2 0.4 0.6 0.8 1
Cc Cυ Correlation function QLC30, CH1, P3, acu101007−220812.raw
[650,690], nIter=0, V0=−0.1581 m/s, nPings=256 dB
−40 −30 −20 −10 0
Figure 5.1 Channel A. Benign channel.
